Measuring the lack of integrabiity of the J_{2} problem for earth's satellites

Abstract

We consider the motion around an oblate primary, keeping only the J2 term in the expansion of the potential in spherical harmonics. The problem has cylindrical symmetry. It has been suspected for a long time, due to numerical evidences, that the problem is non integrable. This has been proved recently [4]. However, even if the system is non integrable, the size of the stochastic zones can be so small that they can be neglected for all practica! purposes. This is what we study here, and we show that for the case of the Earth and considering possible real orbits, i.e., non colliding with the Earth, the effect of the non integrability can be completely neglected.